Disordered RKKY Lattice Mean Field Theory for Ferromagnetism in Diluted Magnetic Semiconductors

We develop a lattice mean field theory for ferromagnetic ordering in diluted magnetic semiconductors by taking into account the spatial fluctuations associated with random disorder in the magnetic impurity locations and the finite mean free path associated with low carrier mobilities. Assuming a carrier-mediated indirect RKKY exchange interaction among the magnetic impurities, we find substantial deviation from the extensively used continuum Zener model Weiss mean field predictions. Our theory allows accurate analytic predictions for $T_c$ and provides simple explanations for a number of observed anomalies, including the non-Brillouin function magnetization curves, the suppressed low-temperature magnetization saturation, and the dependence of $T_c$ on conductivity.